Contact problem and numeric of a planetary drive with small teeth number difference,Shuting Li.pdf
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Contact problem and numeric method of a planetary drivewith small teeth number differenceShuting Li*Nabtesco Co. LTD., Oak-hills No. 202, Heki-cho 7028-2, TSU-shi, Mie-ken 514-1138, JapanReceived 15 July 2007; received in revised form 2 October 2007; accepted 16 October 2007AbstractThis paper deals with a theoretical study on contact problem and numeric analysis of a planetary drive with small teethnumber difference (PDSTD). A mechanics model and finite element method (FEM) solution are presented in this paper toconduct three-dimensional (3D) contact analysis and load calculations of the PDSTD through developing concepts of themathematical programming method T.F. Conry, A. Serireg, A mathematical programming method for design of elasticbodies in contact, Transactions of ASME, Journal of Applied Mechanics 38 (6) (1971) 387392 and finite element methodS. Li, Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear, Transactions ofASME, Journal of Mechanical Design 124 (3) (2002) 511517; S. Li, Finite element analyses for contact strength and bend-ing strength of a pair of spur gears with machining errors, assembly errors and tooth modifications, Mechanism andMachine Theory 42 (1) (2007) 88114 to solve a more complex engineering contact problem. FEM programs are devel-oped through many years efforts. Contact states of teeth, pins and bearing rollers of the PDSTD are made clear throughperforming contact analysis of the PDSTD with the developed FEM programs. It is found that there are only four pairs ofteeth in contact for the PDSTD used as research object when it is loaded with a torque 15 kg m. It is also found that thesefour pairs of teeth are not located in the offset direction of the external gear. They are located at an angular position of 2030? away from the offset direction. Loads shared by teeth, pins and rollers have big difference. The maximum load sharedby the teeth is much greater than the ones shared by pins and rollers. This means that strength calculations of the teeth aremore important than the ones of pins and rollers for the PDSTD. It is also found that all pins share loads while only a partof rollers share loads.? 2007 Elsevier Ltd. All rights reserved.Keywords: Gear; Gear device; Planetary drive; Small teeth number difference; Contact analysis; FEM1. IntroductionIn the latter period of the 20th century, with the development of industry automation, gear devices withlarge reduction ratio found wide applications. Planetary drives with small teeth number difference (PDSTD)was also used widely in automation industry. Though many units of the PDSTD are made every year, strengthdesign calculation of the PDSTD is still a remained problem that has not been solved so far.0094-114X/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2007.10.003*Tel./fax: +81 059 2566213.E-mail address: shutingnpuyahoo.co.jpAvailable online at Mechanism and Machine Theory xxx (2007) xxxxxx/locate/mechmtMechanismandMachine TheoryARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003To perform strength calculations of the PDSTD, it is necessary to know the loads distributed on teeth, pinsand rollers in advance. Since there has not been an effective method available to be able to perform contactanalyses and load calculation of the PDSTD, gear designers have to use ISO standards 46 made for strengthcalculations of a pair of spur and helix gears to perform strength calculations of the PDSTD approximatelyNomenclaturePDSTD planetary drive with small teeth number differenceFEMfinite element methodFEAfinite element analysis3Dthree-dimensionalISOInternational Standard OrganizationFTload on tooth surfaceFPload on pinFRload on rollereeccentricity of the crankshaft.Z1tooth number of the external gearZ2tooth number of the internal gearX1shifting coefficient of the external gearX2shifting coefficient of the internal gearmmodule of gearsB1outside diameter of the internal gearB2inside diameter of the external gearB3diameter of the pin center circle on the external gear(ii0)assumed pair of contact points, also (110), (220), . , (mm0), (aa0), (kk0), (jj0), (bb0), .and (nn0)rused to stand for one elastic body or the external gearsused to stand for the other elastic body or the internal externalekclearance (or backlash) between a optional contact point pair (kk0) before contact. Also, ejFkcontact force between the pair of contact points (kk0) in the direction of its common normalline, also Fjxk, xk0deformations of the assumed pair of contact points (kk0) in the direction of the contact force Fkakj, ak0j0deformation influence coefficients of the contact pointsd0initial minimum clearance between a pair of elastic bodies in the direction of the external forcedrelative displacement of a pair of elastic bodies along the external force under the external force,or angular deformation of the internal gear relative to the external gear under a torque TYslack variables, Y = Y1, Y2, . , Yk, . , YnTXn+1artificial variables, also, Xn+1, Xn+2, Xn+n, . , Xn+n+1Iunit matrix of n n, n is size of the unit matrixZobjective functionSmatrix of the deformation influence coefficientsFarray of contact force of the pairs of contact points, F = F1, F2, . , Fk, . , FnTearray of clearance of the pairs of contact points, e = e1, e2, . , ek, . , enTeunit array, e = 1, 1, .,1, .,1T0zero array, 0 = 0, 0, .,0, .,0Trbradius of the base circle of the internal gearPexternal force applied on a pair of elastic bodiesPGsum of all contact forces between the contact points on tooth surfaces of the PDSTDTtorque transmitted by the PDSTDa0a angle used to express the position of pairs of teeth2S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0037. It has been known that contact problem of the PDSTD is completely different from the one of a pair ofspur and helix gears, so, ISO standards are not suitable for strength calculations of the PDSTD.Manfred and Antoni 8 conducted displacement distributions and stress analysis of a cycloidal drive withFEM. Yang and Blanche 9 also studied design and application guidelines of cycloidal drive with machiningtolerance. Shu 10 conducted study on determination of load-sharing factor of the PDSTD. Chen and Walton11 studied optimum design of the PDSTD.This paper aims to present an effective method to solve contact analysis and load calculation problems ofthe PDSTD. Based on more than 20 years experiences on contact analysis of gear devices and FEM softwaredevelopment, a mechanics model and FEM solution are presented in this paper to conduct contact analysisand load calculations of the PDSTD. Responsive FEM programs are developed through many years efforts.Contact states of the teeth, pins and rollers of the PDSTD are made clear with the developed programs. Loaddistributions on teeth, pins and bearing rollers are also obtained. It is found that there are only four pairs ofteeth in contact for the PDSTD used as research object in this paper when it is loaded under a torque 15 kg m.It is also found that these four pairs of teeth are not located in the offset direction of the external gear.Loads shared by teeth, pins and rollers are compared each other. It is found that the maximum load sharedby teeth is much greater than the ones shared by pins and rollers. It is also found that all pins share loads whileonly a part of rollers share loads. Strength calculations of the PDSTD can be performed easily after loads onteeth, pins and rollers are known.2. Structure and transmission principle introductionsFig. 1 is a simple type of the PDSTD used as research object in this paper. In Fig. 1, this PDSTD consists ofone internal spur gear, one external spur gear, two ball bearings, one input shaft, one output shaft, eight pinsused to transmit torque and 22 rollers used as the center bearing. In order to let teeth of the external gear engagewith the teeth of the internal gear, a radial movement of the external gear relative to the internal gear is needed.This radial movement is realized through rotational movement of a crankshaft. Of course, this crankshaft is acam that can produce offset movement for the external gear (in Fig. 1, when the crankshaft is rotated, a radialmovement of the external gear is produced alternately). The crankshaft is also used as input shaft of the device.Fig. 1 is the position when offset direction of the crankshaft is right up towards to +Y direction. In Fig. 1,O1is the center of the external gear and O2is the center of the internal gear. e is the eccentricity of the crank-shaft. e = O1O2. Gearing parameters and structure parameters of this PDSTD are given in Table 1.Since the PDSTD belongs to K-H-V type of planetary drive and tooth number difference between theinternal spur gear and the external spur gear is small, so this device is often called the planetary drive withsmall teeth number difference. Transmission ratio of this device is equal to Z1/(Z2? Z1) when the internal gearpinsrollersz1z2Input shaftOutput shaftInternal spur gearExternal spur geareAASection A-ACrankshaft (Cam)o1o2pin holeFig. 1. Structure of one kind of planetary drive with small teeth number difference.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx3ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003is fixed. Here, Z1is tooth number of the external gear and Z2is tooth number of the internal gear. From Z1/(Z2? Z1), it can be found that when tooth number difference (Z2? Z1) is very small, transmission ratio Z1/(Z2? Z1) shall become very large. For the device as shown in Fig. 1, teeth number difference (Z2? Z1) isequal to 1, so transmission ratio of this device = Z1.Since an internal gear is used in the PDSTD, tip and root interferences with the mating gear must be checkedlikeausualinternalgeartransmissionwheninvoluteprofileisused.Ofcoursethesetipandrootinterferencescanberemovedthroughperformingtoothprofilemodifications,foranexampletipandrootrelieves.Alsootherpro-files such as modified involute curve, arc profile and trochoidal curves can be used to avoid tip and rootinterferences.3. Load analysis and face-contact model of tooth engagement of the PDSTDFig. 2 is an image of loading state of the external gear in the PDSTD. In Fig. 2, it is found that three kindsof loads are applied on the external gear. They are tooth loads FTproduced by tooth engagement, roller loadsFRproduced in center bearing and pin loads FPresulted from the external torque. Tooth loads are along thedirections of the normal lines of the contact points on tooth surfaces of the internal gear. This also means thetooth contact loads shall be along the directions of the lines of action of the contact points on tooth profile ofthe internal gear. Roller loads are along radial directions of the center hole in the external gear. Pin loads areTable 1Gearing parameters and structural dimensions of the PDSTDGearing parametersGear 1Gear 2Structural dimensionsGear typeExternalInternalDiameter B180 mmTooth numberZ1= 49Z2= 50Diameter B236 mmShifting coefficientX1= 0.0X2= 1.0Diameter B341.125 mmFace width12 mm12 mmPin number8Helical angle00Pin diameter4Module (mm)1Roller number22Pressure angle20?Roller diameter3Tooth profileInvolutesCutter tip radius0.375 mOffset direction+YEccentricity, e0.971 mmRoller loadXYnYYkXkXiTooth loadXnYiPin load123456781234578910612345678910111213141516171819202122Pin center circleFTFRFPFig. 2. Load state of the external gear in the planetary drive.4S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003along the tangential directions of the pin center circle. Though these three kinds of loads are shown in Fig. 2,the reality is that we do not know which tooth, pin and roller shall share or not share loads. This is the prob-lem that must be solved in this paper. Contact analysis with the FEM is presented to solve this problem.Before performing contact analysis of the PDSTD, it is necessary to pay an attention to the tooth engage-ment state of this special device. Tooth engagement of the PDSTD is different from a usual internal gear trans-mission in that tooth engagement of a usual internal gear transmission is an engagement of teeth on thegeometrical contact lines and it has been already known in theory how many teeth and which teeth shall comeinto contact in different engagement positions for the usual internal gear transmission while tooth engagementof the PDSTD is not on the geometrical contact lines and it is not known in theory where the teeth shall con-tact on tooth profile, how many teeth shall come into contact and which teeth shall come into contact for thePDSTD. Even, it is not known whether the geometric contact lines exits or not for the PDSTD.The other difference is contact state of one pair of teeth. As it has been stated above, for a usual internalgear transmission, a pair of teeth shall contact on the geometrical contact line. It is called Line-contact of atooth in this paper. But for the PDSTD, the teeth shall contact on a part of face on the profile like the har-monic drive device. It is called Face-contact of a tooth in this paper. Fig. 3 is the real tooth contact states ofthe PDSTD with the parameters as shown in Table 1. From Fig. 3, it is found that the teeth 5, 6, 7, 8 and 9 areface-contact on the most part of tooth profile. So when to perform contact analysis of loaded teeth of thePDSTD with the FEM, a lot of pairs of contact points (ii0), (jj0), (kk0) and (nn0) as shown in Fig. 4 mustbe made between the tooth profiles of the external and the internal gears. These pairs of contact points areassumed to be in contact at first and it shall be made clear finally which pair of points turns out not to bein contact through performing contact analysis of the PDSTD with the FEM presented in this paper.4. Basic principle of elastic contact theory used for contact analysis of a pair of elastic bodies 14.1. Deformation compatibility relationship of a pair of elastic bodiesIn Fig. 5,randsare one pair of elastic bodies which will come into contact each other when an externalforce P is applied. The contact problem to be discussed here is restricted to normal surface loading conditions.InternalgearExternalgear56789Fig. 3. Face-contact of mating teeth.Internal gearExternal gearijknijknFjjFkkFig. 4. Pairs of contact points on tooth surfaces of the internal gear and the external gear.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx5ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003Discrete forces can be taken to represent distributed pressures over finite areas. The following assumptions aremade: (1) deformations are small; (2) two bodies obey the laws of linear elasticity; and (3) contact surfaces aresmooth and have continuous first derivatives. With above assumptions, contact analysis of this pair of elasticbodies can be made within the limits of the elasticity theory.In Fig. 5, contact of this pair of elastic bodies is handled as contact of many pairs of points on both sup-posed contact surfaces ofrandslike gears contact as shown in Fig. 4. These pairs of contact points arePP123makjbqnk123makjbqn0Supposedcontact faceFjFjFjFj(a) Three-dimensional view PPakjbakjbakjbakjbkkkBefore contactAfter contact(b) Section view Fig. 5. Model of a pair of elastic bodies: (a) three-dimensional view and (b) section view.6S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003expressed as (110), (220), . , (mm0), (aa0), (kk0), (j-j0), (bb0), . and (nn0). n is the total number of con-tact point pairs assumed. Fig. 5b is a section view of Fig. 5a in the normal plane of the contact bodies. InFig. 5, ekis a clearance (or backlash) between a optional contact point pair (kk0) before contact. Fkis contactforce between the pair of contact points (kk0) in the direction of its common normal line when k contacts withk0under the load P (It is assumed that all the common normal lines of the contact point pairs are approxi-mately along the same direction of the external force P in this paper because a contact area is usually verynarrow. This assumption is reasonable in engineering, but we shall use the real direction of the contact pointpairs in this paper). xk, xk0are deformations of the points k and k0in the direction of the force Fkafter con-tact. d0is the initial minimum clearance betweenrandsand d is displacement of the points O1relative tothe point O2(the loading points in Fig. 5b).For the optional contact point pair (kk0), if (kk0) contacts, (xk xk0 ek), the amount of the deforma-tions and clearance on the point pair (kk0), shall be equal to the relative displacement quantity d, and if (kk0)does not contact, (xk xk0 ek) shall be greater than d. Eqs. (1) and (2) can be used to express these relation-ships in the following. Eq. (3) is used to sum Eqs. (1) and (2):xk xk0 ek? d 0Not contact1xk xk0 ek? d 0Contact2Then,xk xk0 ek? d P 0k 1;2;.;n3According to Hertzs theory, contact deformation under the external force P has the relationship with outlinesof the contact surfaces and the external force P. This means the contact deformation is determined by twofactors, geometry of the contact surfaces and the external force P. When the external force P is changed, con-tact area of a pair of elastic bodies is also changed correspond. This change of the contact area makes it a non-linearity, the relationship between contact deformation and the external force P. But since this non-linearity isonly resulted from increase and decrease in contact areas, this non-linearity is the so-called geometric non-linearity, not the so-called material non-linearity. So, for the pairs of points assumed to be in contact, rela-tionship between deformation and contact force (force on contact point pairs, not the external force P) is stilllinearity when elastic deformations are considered. Then the elastic deformations xkand xk0of the pairs ofpoints in contact can be expressed with Eq. (4) by using deformation influence coefficients akjand ak0j0,xkXnj1akjFj;xk0Xnj1ak0j0Fj4where Fjis contact force between the point pair (jj0). If Eq. (4) is substituted into Eq. (3), (5) can be obtainedand if Eq. (5) is expressed in a form of matrix expression, Eq. (6) can be obtained,Xnj1akj ak0j0Fj ek? d P 05S?fFg feg ? dfeg P f0g6whereS? Skj? akj ak0j0?fFg fF1;F2;.;Fk;.;FngTfeg fe1;e2;.;ek;.;engTfeg f1;1;.;1;.;1gTf0g f0;0;.;0;.;0gTk;j 1;2;.;n; k0;j0 10;20;.;n0S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx7ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0034.2. Force equilibrium relationship of elastic contact bodiesInthispaper,itisassumedthatallthecontactforcesbetweenthecontactpointpairsarealongthedirectionoftheexternalforceP.Sinceacontactareaisusuallyverysmall,thisassumptionisreasonableinengineering.Withtheassumption,itcanbethoughtthattheexternalforcePisequaltothesumofallthecontactforceFj(j = 1ton).Then Eq. (7) can be obtained. If the Eq. (7) is written in a form of matrix expression, Eq. (8) can be obtained,P Xnj1Fj7fegTfFg P84.3. Mathematical programming method used to calculate contact loadsEqs. (6) and (8) are constrain conditions to identify which point pair contacts or not. Contact problem ofthe pair of elastic bodiesrandscan be stated as looking for the contact load Fj(j = 1, 2, 3, . , n) thatmeets the constraint equations (6) and (8) under the condition of knowing the deformation influence coeffi-cients akj, ak0j0, clearance ekand the external force P in advance. But only with these two constraint equations,it is impossible to find the contact load Fj. This is because there is not a mathematical method available to beable to treat this problem that only has two constraint equations and no objective function.The above-stated problem is very like the problems that the Modified Simplex Method of the mathematicalprogramming principle can treat. According to the principle of the Modified Simplex Method, the problemwith only constraint equations and no objective function can be treated as a mathematical programmingmodel by building an artificial objective function through introducing some positive variables.So, the Modified Simplex Method of the mathematical programming method is used here to build a math-ematical programming model and solve the contact analysis problem of the pair of elastic bodies.Since Eq. (6) is an inequality constraint equation that may be strictly positive or identically zero, it shouldbe transformed into an equality constraint equation by introducing a so-called slack variable Y (a positivevariable) according to the principle of the Modified Simplex Method. Then Eqs. (9) and (10) can be obtained,S?fFg feg ? dfeg ? I?fYg f0g9or?S?fFg dfeg I?fYg feg10where Y = Y1Y2. Yk. YnT(slack variables, YkP 0, k = 1, 2, . , n) and I is the unit matrix of n n.Objective function Z is introduced according to the principle of the Modified Simplex Method throughintroducing some positive variables Xn+1, Xn+2, . Xn+n, Xn+n+1(usually called artificial variables). Thenthe mathematical model of the Modified Simplex Method for contact analysis of the pair of elastic bodiescan be built as following Eqs. (11)(13) based on the principle of the Modified Simplex Method 1214.Objective functionZ Xn1 Xn2 . Xnn Xnn111Constraint conditions? S?fF? dfeg I?fYg I?fZ0g feg12fegTfFg Xnn1 P13whereS? Skj? akj ak0j0?;k 1;2;.;n; j 1;2;.;nfZ0g fXn1;Xn2;.;XnngTfFg fF1;F2;.;Fk;.;FngTfYg fY1;Y2;.;Yk;.;YngT8S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003feg fe1;e2;.;ek;.;engTFkP 0;YkP 0;ekP 0;d P 0;k 1;2;.;nXnmP 0;m 1;2;.;n 1Eq. (11) is the objective function introduced based on the principle of the Modified Simplex Method. Eq. (12)is the one constraint equation obtained from the Eqs. (10). In Eq. (12), IZ0 is also introduced according tothe principle of the Modified Simplex Method. Eq. (13) is the other constraint equation obtained from Eq. (8).In Eq. (13), Xn+n+1is also introduced according to the principle of the Modified Simplex Method.With the above mathematical model of the mathematical programming method, contact analysis problemof the pair of elastic bodies can be stated as performing mathematical programming with the objective func-tion of Eq. (11) under the two constraint equations (12) and (13). Or say it in more detail, finding the contactload F under the condition of knowing the deformation influence coefficient matrix S, clearance array eand the external force P in advance through performing mathematical programming using the objective func-tion of Eq. (11) and two constraint equations of Eqs. (12) and 13. The above mathematical model is a standardform of the mathematical programming method, so, Eqs. (11)(13) can be solved simply by the Modified Sim-plex Method 1214.5. FEM for contact analysis and load calculations of the PDSTD5.1. Mechanics model used for contact analysisMethods stated in Section 4 cannot be used directly to solve contact problem of the PDSTD. This isbecause involute curve is used as tooth profile of the PDSTD and different contact points on tooth profilesas shown in Fig. 4 have different load directions. So, if we only think that contact analysis of a pair of elasticbodies must be conducted in the one same load direction as shown in Fig. 5, the above methods can not beused for contact analysis of the PDSTD.But if we change our mind and develop concepts of the above methods from one load direction contact toall load direction contacts of the contact points, then basic principle in Section 4 can be developed to solvecontact problem of the PDSTD. The following is an introduction to show how to develop the basic principlein Section 4 to solve contact problem of the PDSTD.Fig. 6a is the mechanics model used for contact analysis of the PDSTD. In Fig. 6a, all rollers are replacedwith supports at the contact points. These supports are called roller supports and they can only carry loadsalong radial directions and are free in the tangential directions. Also, all pins are replaced with pin supports.These pin supports can only carry loads in tangential directions of the pin center circle and are free in theradial directions of the pin center circle.In Fig. 6a, if an external torque T is applied on the internal gear, tooth loads on the pairs of contact pointson tooth surfaces shall be produced. Fig. 6b is used to show positions of the contact points and load directionsof the contact points on loaded tooth surfaces of the internal gear. From Fig. 6b, it is found that differentcontact points at different positions have different load directions, but since involute curve is used as toothprofile, all the load directions of the contact points at different positions shall be along their lines of actionat their positions. This means that all the normal lines of the contact points shall be tangential to the samebase circle of the internal gear as shown in Fig. 6b. So, the following equation can be obtained,T Xnj1Fj? rb14where Fjis tooth load on an arbitrary contact point j as shown in Fig. 6b. n is total number of all the contactpoints on all the loaded tooth surfaces of the internal gear. rbis radius of the base circle of the internal gear.Eq. (14) can also be rewritten as Eq. (15) since rbis a constant,T rb?Xnj1Fj15S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx9ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003Fjis in Eq. (15) has different meaning with the Fjin Eq. (7) in that the former has different load directions whilethe latter has the same load direction.In Fig. 6a, boundary support points used to stand for pins and bearing rollers shall receive reaction forcesresulted from tooth loads in order to make a balance with the tooth loads. Since rollers can only receive radialloads and pins can only receive tangential loads, boundary support points are made as shown in Fig. 6a. Then,contact problem of the PDSTD can be stated as follows.When a torque T is loaded on the internal gear of the PDSTD, contact points on tooth surfaces shall havecontact loads along the lines of action of themselves, boundary support points on the center hole of the exter-o2o1XYeT(a) Mechanics modelContact pointsBase circle ofinternal gearLines of actionjFjrb(b) Contact points and lines of action of the contact points Fig. 6. Section view of the mechanics model used for FEM contact analysis.10S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003nal gear shall have contact loads along radial directions of the center hole and boundary support points usedto stand for pins shall have contact loads along the tangential directions of the pin center circle. Contact anal-ysis of the PDSTD is changed into loaded tooth contact analysis of the PDSTD along all the load directions ofcontact points on tooth surfaces under the boundary conditions that pins and rollers are supported. Of course,the boundary conditions are not known when the loaded tooth contact analysis is conducted. This specialloaded tooth contact analysis with uncertain boundary conditions is realized through performing iteration cal-culations of FEM and mathematical programming method.5.2. Assuming pairs of contact teeth and pairs of contact pointsSince it is not known in theory in advance how many pairs of teeth are in contact, which pair of the teeth isin or not in contact for the PDSTD, a large range of pairs of teeth are assumed to be in contact in advancewhen a torque T is loaded. In Fig. 6a, about half of the internal gear is used as FEM model for the contactanalysis of the PDSTD. This means that all the internal gear teeth used as the model as shown in Fig. 6a areassumed to be in contact with teeth of the external gear. The whole external gear is used as FEM model for thecontact analysis of PDSTD.On the other hand, many pairs of contact points on tooth surfaces of every assumed pair of contact teethare made like the pairs of contact points as shown in Fig. 4. In Fig. 4, only four pairs of the contact points (ii0), (jj0), (kk0), (nn0) are given as an example. For a real calculation, it is necessary to make enough pairs ofcontact points in order to get correct tooth load distributions. Thirteen pairs of contact points are made alongtooth profile for every assumed pair of contact teeth in this paper when the real contact analysis is performed.In Fig. 4, pairs of contact points are made along their lines of action of themselves. For an example, the pairof contact points (jj0) is made along direction of the line of action at the point j0and the pair of contact points(kk0) is made along direction of the line of action at the point k0. Of course, all the pairs of the contact pointshave different directions of the lines of action.5.3. How to determine boundary conditions to the external and internal gearsIn Fig. 1, the internal gear is often fixed on motor flange through bolts. So, when the partial tooth FEMmodel as shown in Fig. 7b is used to calculate deformation influence coefficients of the internal gear with 3D,FEM, nodes on the outside circumferential face and two end faces of the partial tooth model as shown inFig. 7a and b are fixed in three directions as boundary conditions in FEA. This boundary is not changedin all the calculations for the internal gear.For the case of the external gear, as it is shown in Fig. 8c, the external gear is supported by rollers and pins.The rollers can only support the external gear in radial directions and the pins can only support the externalgear in circumferential direction. Before FEA of the external gear, it is not known which pin and roller willsupport the gear and which will not. So, at the first time of FEA, it is assumed that all the pins will support theexternal gear only in circumferential direction and all the rollers shall support the external gear only in radialdirections. Then FEA for the external gear can be conducted and reaction forces on these pins and rollers canbe available after FEA. The next task is to identify directions of the reaction forces on these pins and rollers. Ifthe reaction forces are tensile load, this means that these pins and rollers with tensile loads should be free in thenext a calculation. On the other hand, if the reaction forces are compressive loads, these pins and rollers withthe compressive loads should be fixed continually in the next a calculation. In this way, FEA is repeated untilno load direction changes for all the pins and rollers. Then the correct boundary conditions for pins and roll-ers are obtained. The correct boundary conditions are used finally in FEA of the external gear.Fig. 8b is an image of the boundary nodes used to stand for the pins and rollers.5.4. Calculating deformation influence coefficients and gaps of all the assumed pairs of contact pointsWhen all pairs of contact teeth are assumed and all pairs of contact points on all the assumed pairs of con-tact teeth are made, the next task is to calculate deformation influence coefficients and gaps of all the assumedS. Li / Mechanism and Machine Theory xxx (2007) xxxxxx11ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003pairs of contact points on tooth surfaces of the external and the internal gears along the lines of action ofthemselves.Gaps of all the assumed pairs of contact points can be calculated geometrically when the positions of thepairs of contact points are determined artificially. As shown in Fig. 4, ejis gap of the pair of contact points (jj0) and ekis gap of the pair of contact points (kk0). Directions of ejand ekare different. They are calculatedalong the lines of action of themselves.Deformation influence coefficients of the contact points on tooth surface of the external and the internalgears are calculated separately with 3D, FEM. As stated above, deformation influence coefficients of differentcontact points have different directions. They are calculated along their lines of action of themselves. For anexample, in Fig. 4, Deformation influence coefficient of the point j0on tooth surface of the internal gear iscalculated along the line of action at the point j0while deformation influence coefficient of the point k0ontooth surface of the internal gear is calculated along the line of action at the point k0. This is the developmentof a traditional mathematical programming method from in one direction contact to multi-directions contact.Calculation of deformation influence coefficients of the contact points on tooth surface of the internal gearis simpler than the external gear. FEM model as shown in Fig. 7b is used for deformation influence coefficientcalculations of the internal gear. When deformation influence coefficients are calculated by 3D, FEM, bound-ary nodes as shown in Fig. 7a are fixed. Since boundary conditions of the internal gear are not changed, it isnot necessary to calculate the deformation influence coefficients again. Fig. 7a is a section view to show theboundary conditions of the internal gear as shown in Fig. 7b. All the boundary nodes on two end facesand outside cylinder face of the model as shown in Fig. 7b are fixed as boundary conditions when the defor-mation influence coefficients are calculated.Deformation influence coefficient calculation of the contact points on tooth surface of the external gear ismore complex than the internal gear. Since it is unknown in advance in theory which pin and roller shall be inor not in contact, boundary conditions cannot be fixed in calculation of the deformation influence coefficientsNodes used as boundary fixed points for FEA(a) Nodes used as boundary conditions (b) FEM mesh-dividing pattern of the internal gear FixedFixedFixedFig. 7. FEM model used to calculate deformation influence coefficients of the contact points on tooth surfaces of the internal gear.12S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003of the contact points on the external gear with 3D, FEM. FEM calculations must be repeated as follows inorder to get correct deformation influence coefficients of the contact points on tooth surfaces of the externalgear.When deformation influence coefficients of the contact points on tooth surfaces of the external spur gear arecalculated, FEM model as shown in Fig. 8 is used. At the first time of the calculation, all pin supports androller supports are fixed (nodes as shown in Fig. 8b are fixed) as the boundary condition in the directionsas shown in Fig. 8c and deformation influence coefficients of all the assumed contact points are calculatedCamPinsRollersNodes used as boundary fixed points for FEAabco1XdFig. 8. FEM model used to calculate deformation influence coefficients of the contact points on tooth surfaces of the external gear.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx13ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003by 3D, FEM. Of course, it is possible enough that some pins and rollers turn out not to be in contact. Thismeans that some wrong boundary conditions are given at the first time. But since new boundary conditionscan be gotten after contact analysis, calculations of the deformation influence coefficients shall be repeatedunder the new boundary conditions until correct deformation influence coefficients can be obtained under cor-rect boundary conditions.5.5. Calculating the external force PFrom Eq. (15), Eq. (16) can be obtained.PG T=rbXnj1Fj16Here, PGhas different meaning from the P in Eq. (7). PGis sum of values of all the contact loads with differentdirections on contact points. Load directions are not included in PG, here.5.6. Calculating tooth loads distributed on all the assumed pairs of contact pointsEqs. (11)(13) can be used directly to conduct contact analysis of the PDSTD without any changes. Theonly difference is that deformation influence coefficients akj, ak0j0gap e and contact load Fjare all calculatedalong their lines of action of the contact points.When akj, ak0j0, e and P are known, tooth loads F and d can be calculated by solving the Eqs. (11)(13)with the Modified Simplex Method of mathematical programming principle. Here, d means angular deforma-tion of the internal gear relative to the external gear. Tooth contact pattern and load-sharing rate of everytooth can be calculated based on the values of F. Friction between the contact tooth surfaces is ignoredin all the calculations, since gear transmission has a high efficiency.5.7. Pin and roller load distribution calculationsAfter tooth load distributions are known, models as shown in Figs. 2 and 8 are used to calculate pin androller loads through performing FEA of the external gear. Firstly, tooth loads are applied on the loaded toothsurfaces, then pin and roller loads are calculated through performing FEA and calculating reaction forces onthe fixed boundary nodes (pin and roller supports as shown in Fig. 8).Contact states of the pins and rollers can be known when the reaction forces on the fixed boundary nodesare known. If the reaction force on a pin or roller is a tensile force, this means that this pin or roller cannotcarry load and this pin or roller should be free in the next one calculation. If the reaction force on the pin orroller is a compressive force, this means that this pin or roller carries load and this pin or roller should be fixedcontinually in the next one calculation. Like this, new boundary conditions of the pin and roller contact statesare formed and they are feedback to Section 5.3 for the next one calculation of the deformation influence coef-ficients of the assumed contact points on tooth surfaces of the external gear.5.8. Next one calculationGo back to Section 5.3 and repeat the calculation procedures from the Sections 5.35.7 until contact statesof the pins and rollers have no more changes comparing with the last one calculation. Then output tooth, pinand roller loads.5.9. FEM software developmentFEM software is developed through many years efforts with Visual Fortran language in a personal com-puter. Super-parametric hexahedron solid element 2,3 is used. The following is flowchart of the FEM soft-ware development. Every step in the flowchart is also explained in the following.14S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003Step 1: Input gearing parameters and structure dimensions of a pair of gearsThe FEM software is built to be able to conduct contact analysis and load calculations of a PDSTD witharbitrary structure dimensions and gearing parameters as shown in Table 1. It can also be used to analyze thegears with non-standard (long or short) addendum when addendum coefficient is inputted.Step 2: Input FEM mesh-dividing parameters, tooth engagement position parameter and external torqueEngagement position of a tooth from engaging-in to engaging-out is expressed as a parameter in the paper.When value of this parameter is given continually, engagement position of the tooth from engaging-in toengaging-out can be determined continually. Of course, this parameter must be given before calculation.FEM mesh-dividing pattern of the external and internal gears are also controlled by other parameters. Withthese parameters, it can be determined where should be fine divided and where should be roughly divided.Usually, tooth contact areas and tooth root are fine divided. FEM mesh-dividing pattern of the gears canbe changed simply through changing values of these parameters.Transmission torque is also given here for contact analysis.Step 3: Divide FEM meshes of the external gear and the internal gear automaticallyPrograms are developed to be able to divide FEM meshes of the external gear and the internal gear auto-matically when structure dimensions, gearing parameters, tooth engagement position parameter and FEMmesh-dividing parameters are given.Step 4: Give tooth contact range of the internal gear and form pairs of contact points on tooth surfaces ofthe external and internal gearsTooth contact range of the internal gear as shown in Fig. 7 is given here. Pairs of contact points on toothsurfaces of the external and internal gears as shown in Fig. 4 are also made automatically here.Step 5: Calculate backlash ekof every pair of contact pointsBacklash ekof every pair of contact points is calculated geometrically and automatically.Step 6: Give boundary conditions to the external gear and the internal gearNodes as shown in Fig. 8b are fixed as boundary conditions for deformation influence coefficient calcula-tions of the external gear and nodes as shown in Fig. 7a are fixed as boundary conditions for deformationinfluence coefficient calculations of the internal gear.Step 7: Calculate deformation influence coefficients of the contact points on the tooth surfaces by 3D, FEMWhen the pairs of contact points on tooth surfaces of the external gear and the internal gear are made, defor-mation influence coefficients of all the assumed contact points are calculated by 3D, FEM. For an example,when to calculate deflection influence coefficients of a contact point on tooth surface of the external gear,FEM model as shown in Fig. 8 is used and a unit force is applied on the contact point on tooth surface alongthe line of actionof itself, then deformations of the contact point and other contact points on its and other toothsurfaces are calculated with a 3D, FEM along the lines of action of themselves. This calculation is repeated forall the remained contact points on all the tooth surfaces. Then deformation influence coefficient matrix of all thecontact points on all the tooth surfaces of the external gear is formed by assembling these calculated deforma-tionsintoamatrix.ThesamecalculationsaremadefortheinternalgearbyusingFEMmodelasshowninFig.7.Step 8: Set up Eqs. (11)(13) and solve these equations to get tooth loadsSet up mathematical model of Eqs. (11)(13) and solve these equations with the Modified Simplex Methodof the mathematical programming method. Tooth loads (contact forces between the pairs of contact points)are obtained here.Step 9: Calculate loads on pins and rollers by FEALoads on pins and rollers are calculated by FEA through calculating reaction forces on the fixed nodes onboundary of the external gear.Step 10: Form new boundary conditions based on directions of the calculated pin and roller loadsIn Step 6, all pin supports and roller supports are fixed as the boundary conditions for deformation influ-ence coefficient calculations of external gear. These boundary conditions are not correct enough at the firsttime. The correct boundary conditions can be formed based on the load directions on the pin supports androller supports. If the load is calculated to be a compressive load on a pin or a roller, this means, this pinor roller shall be fixed continually in the next calculation. On the other hand, if the load is calculated to bea tensile load on a pin or a roller, this means, this pin or roller cannot carry load and this pin or roller shouldbe released (free) in the next calculation. After all load directions on pins and rollers are checked like this, theS. Li / Mechanism and Machine Theory xxx (2007) xxxxxx15ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003new boundary conditions can be formed. This new boundary conditions shall be used in Step 7 for deforma-tion influence coefficient calculations of the external gear in the next calculation.Step 11: Are the boundary conditions changed?New boundary conditions of the external gear obtained in Step 10 are compared with the old conditionsgiven in the last calculation. If no changes happen, then go to Step 12. If there are changes for the boundaryconditions, then go to Step 7 to do a new calculation with the new boundary conditions. The old boundaryconditions means the boundary conditions obtained in the previous calculation and the new boundary condi-tions means the boundary conditions obtained at the current calculation.Step 12: Output tooth loads, pin loads and roller loadsTooth loads, pin loads and roller loads are outputted here.Step 13: StopInput gearingparametersandstructuredimensionsofapairofgearsDivideFEMmeshof the internal gearautomaticallyInput FEMmesh-dividingparameters, toothengagementpositionparameterandexternal torqueTGive toothcontact rangeof the internal gearandformpairsofcontact pointson toothsurfacesof theexternal and internal gears.DivideFEMmeshof theexternal gearautomaticallycalculatebacklash kof every pair of contact pointsCalculatedeformation influencecoefficientsof thecontact pointson toothsurfacesof theexternal gearby3D, FEMCalculatedeformation influencecoefficientsof thecontact pointson toothsurfacesof theinternal gearby3D, FEMSet upequations(11), (12)and(13)andsolve them toget tooth loadsCalculate loadsonpinsandrollersbyFEAFormnewboundaryconditionsbasedondirectionsof thecalculated loadsonpinsandrollersSTOPGiveboundaryconditionstotheexternalgearGiveboundaryconditions to the internal gearAre theboundaryconditionschanged?(Comparingwith the last calculation)Output loadson teeth, pinsandrollersNoYes16S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0036. An example of calculationsGearing parameters and structure parameters are given in Table 1. Offset direction of the external gear istoward +Y-axis as shown in Figs. 1, 2 and 6 in all the calculations.Fig. 9a is 3D, FEM model used in the calculations. Fig. 9b is a section view of Fig. 9a and only a part ofteeth are shown in Fig. 9b. In Fig. 9, engaged teeth are fine divided. Sixteen meshes (17 nodes) are made withinface width, 12 meshes are made within tooth profile and 8 meshes are made within fillet part of tooth root.Output torque is 15 kg m in all the calculations. Calculations results are given in the following.Fig. 9. 3D, FEM Model for contact analysis of the PDSTD.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx17ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0036.1. Tooth load distributionFig. 10 is a relationship between the minimum backlash of every pair of teeth and position of the pair ofteeth. In Fig. 10, the position of the pair of teeth is expressed as a angle a0as shown in Fig. 2. Total load onevery pair of mating teeth is calculated based on all the node loads obtained, then relationship between thetotal load and position of the pair of teeth is also given in Fig. 10. Tooth number as shown in Fig. 2 is alsogiven in Fig. 10. From Fig. 10, it is found that only the four teeth (teeth 5, 6, 7 and 8) share tooth loads andother teeth do not share loads. From Fig. 2, it is found that teeth 5, 6, 7 and 8 are not located on the upside ofthe offset direction (a0= 90? in Fig. 2) of the external gear. They are on the left side of the Y-axis. This isbecause the teeth 4, 5, 6, 7 and 8 have smaller backlash of mating teeth. In Fig. 10, it is also found thatthe total load on teeth 6 and 7 are much greater than the teeth 5 and 8.Node loads are summed along the direction of tooth profile of a tooth. Fig. 11 is the longitudinal distribu-tion of this summed tooth load. From Fig. 11, it is found that the longitudinal distribution of the tooth load isnot a straight-line parallel to the abscissa. This is because pins are supported on only the left side as shown inFig. 8a and b. As it is shown in Fig. 8a, a torque is transmitted with the pins on the left side of the gear. Soonly pins (or nodes) on the left side of the gear in Fig. 8b are fixed as boundary conditions in FEA. If the pinsinserted in the gear are not like a cantilever structure, they are supported on two sides of the external gear,Fig. 11 shall become symmetric distribution relative to the center of face width.Fig. 12a is contour lines of the calculated contact point loads on the surface of Tooth 5. Fig. 12bd are thesame results on surfaces of teeth 6, 7 and 8 separately. In Fig. 12, the abscissa is the face width of the teeth andthe ordinate is the position of points along tooth profile. The ordinate = 1 is the point at tooth tip and the607080901001101201301401501600.00.4Angular position of mating teeth deg.Toot backlash mm Backlash050010001500200025008765923456781Torque=15kgmTeeth on theleft sideTeeth on theright sideOffset directionTooth load N Tooth loadFig. 10. Tooth backlash and tooth load distribution.024681012020406080100120140160180 Tooth 8 Tooth 5 Tooth 7 Tooth 6Total load along profile NFace width mmFig. 11. Longitudinal distribution of the tooth load.18S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003ordinate = 13 is the point at tooth root. So Figs. 12ad are tooth loads distributed on entire the tooth surfacesof the teeth 5, 6, 7 and 8 separately.From Fig. 12a and b, it is found that the teeth 5 and 6 are contacted on the areas of tooth tip while the teeth7 and 8 are contacted on the middle part of the teeth.6.2. Pin load distributionFig. 13 is a relationship between pin load and pin position. In Fig. 13, pin position is also expressed as theangle a0as shown in Fig. 2 and the ordinate is pin load calculated. Pin number as shown in Fig. 2 is also givenin Fig. 13. From Fig. 13, it is found that all eight pins share loads and the maximum load is on Pin 4.6.3. Roller load distributionFig. 14 is a relationship between the total load on roller and roller position. In Fig. 14, roller position is alsoexpressed as the angle a0as shown in Fig. 2 and the ordinate is total load on roller. Roller number as shown inFig. 2 is also given in Fig. 14. From Fig. 14, it is found that only rollers 1, 2, 3, 4, 5, 6, 19, 20, 21 and 22 share2.04.06.08.010121410122024684681012Unit: NLoad distribution on entire the tooth surfaceRootTipTooth 5Position of points on tooth profile Face width mm101202468Face width mm101202468Face width mm101202468Face width mm(a) Loads on the surface of Tooth 5 1114141720222524681012Unit: NLoad distribution on entire the tooth surfaceRootTipTooth 6Position of points on tooth profile(b) Loads on the surface of Tooth 6 9.0141418182327313624681012Unit: NLoad distribution on entire the tooth surfaceRootTipTooth 7Position of points on tooth profile (c) Loads on the surface of Tooth 7 8.09.61124681012Unit: NLoad distribution on entire the tooth surfaceRootTipTooth 8Position of points on tooth profile(d) Loads on the surface of Tooth 8 Fig. 12. 3D distribution of tooth load on tooth surface.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx19ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003loads and the remained rollers do not share loads. From Fig. 2, it is found that the rollers 1, 2, 3, 4, 5, 6, 19, 20,21 and 22 are all located on the right side of the Y-axis.Fig. 15 is the longitudinal distribution of roller load. From Fig. 15, it is found that Roller 5 has the greatestload.0306090120 150 180 210 240 270
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